# Help with Anti-derivative problem

• May 19th 2008, 01:47 AM
CALsunshine
Help with Anti-derivative problem
Find f.

f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1

Any insight as to how to handle this would be wonderful :)
• May 19th 2008, 03:00 AM
mr fantastic
Quote:

Originally Posted by CALsunshine
Find f.

f ' (x)= 4/{square root(1-2[squared])}, f(1/2)= 1

Any insight as to how to handle this would be wonderful :)

Is f ' (x) meant to depend on x at all .......?
• May 19th 2008, 04:09 AM
Soroban
Hello, CALsunshine!

I assume there is a typo . . .

Quote:

$f'(x)\:=\:\frac{4}{\sqrt{1-x^2}},\quad f\left(\frac{1}{2}\right)\:=\: 1$

Find $f(x).$

Integrate: . $f(x) \;=\;\int\frac{4}{\sqrt{1-x^2}}\,dx \quad\Rightarrow\quad f(x) \;=\;4\sin^{\text{-}1}(x) + C$

Since $f\left(\frac{1}{2}\right) \,=\,1$, we have: . $1 \;=\;4\sin^{\text{-}1}\left(\frac{1}{2}\right) + C$

. . Then: . $1 \;=\;4\left(\frac{\pi}{6}\right) + C \quad\Rightarrow\quad C \:=\:1 - \frac{2\pi}{3}$

Therefore: . $f(x)\;=\;4\sin^{\text{-}1}(x) + 1 - \frac{2\pi}{3}$